Linear Inequalities
5.1 Introduction
Linear inequalities are mathematical expressions that show the relationship between two expressions using inequality symbols.
Key Concepts:
- Inequality symbols: <, >, ≤, ≥, ≠
- Solutions represent ranges of values rather than specific points
- Applications in optimization, economics, and engineering
5.2 Inequalities
Types of Inequalities:
| Type | Example | Solution Interpretation |
|---|---|---|
| Strict | x > 3 | All numbers greater than 3 |
| Non-strict | x ≤ -2 | All numbers less than or equal to -2 |
| Compound | 1 < x ≤ 5 | Numbers between 1 and 5, including 5 |
Properties of Inequalities:
- Addition/Subtraction: If a > b, then a ± c > b ± c
- Multiplication:
- If c > 0 and a > b, then ac > bc
- If c < 0 and a > b, then ac < bc (sign reverses)
- Transitive: If a > b and b > c, then a > c
Example:
Solve: 2x + 5 > 11
2x > 11 - 5
2x > 6
x > 3
5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
Solving Linear Inequalities:
- Isolate the variable term on one side
- Simplify the inequality
- Remember to reverse the inequality sign when multiplying/dividing by negative numbers
Number Line Representation:
For x > 2:
2 x > 2
Key:
- Open circle (○): > or <
- Closed circle (●): ≥ or ≤
- Arrow: Direction of solution set
Graphical Representation of Inequalities:
For y ≤ 2x + 1:
[Graph would show solid line y=2x+1 with shaded area below]
Key:
- Solid line: ≤ or ≥ (inclusive)
- Dashed line: < or > (exclusive)
- Shaded region: Solution area
Example:
Solve and graph: -3x + 7 ≥ 4
-3x ≥ -3
x ≤ 1 (sign reversed when dividing by -3)
Number line representation:
1 x ≤ 1
Applications:
- Budget constraints in economics
- Range of acceptable measurements in engineering
- Feasible regions in optimization problems